Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 637209, 15 pages
doi:10.1155/2012/637209
Research Article
The Second-Order Born Approximation in
Diffuse Optical Tomography
Kiwoon Kwon
Department of Mathematics, Dongguk University, Seoul 100715, Republic of Korea
Correspondence should be addressed to Kiwoon Kwon, kwkwon@dongguk.edu
Received 21 October 2011; Accepted 8 December 2011
Academic Editor: Chang-Hwan Im
Copyright q 2012 Kiwoon Kwon. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
Diffuse optical tomography is used to find the optical parameters of a turbid medium with infrared
red light. The problem is mathematically formulated as a nonlinear problem to find the solution
for the diffusion operator mapping the optical coefficients to the photon density distribution on
the boundary of the region of interest, which is also represented by the Born expansion with
respect to the unperturbed photon densities and perturbed optical coefficients. We suggest a new
method of finding the solution by using the second-order Born approximation of the operator.
The error analysis for the suggested method based on the second-order Born approximation is
presented and compared with the conventional linearized method based on the first-order Born
approximation. The suggested method has better convergence order than the linearized method,
and this is verified in the numerical implementation.
1. Introduction
Diffuse optical tomography involves the reconstruction of the spatially varying optical properties of a turbid medium. It is usually formulated as inverse problem with respect to the forward problem describing photon propagation in the tissue for given optical coefficients 1.
The forward model is described by the photon diffusion equation with the Robin
boundary condition. In the frequency domain, it is given by
iω
Φ
q
−∇ · κ∇Φ μa c
Φ 2aν · κ∇Φ 0
in Ω,
1.1
on ∂ Ω,
where Ω is a Lipschitz domain in Rn , n 2, 3, . . ., ∂Ω is its boundary, ν is the unit outward
normal vector on the boundary, Φ is the photon density, q is a source term, a is a refraction
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parameter, and μa , μs , and κ 1/3μa μs are the absorption, reduced scattering, and
diffusion coefficients, respectively. Assume that a is a constant and κ, μa , μs are scalar functions satisfying
0 < L ≤ κ, μa , μs , a ≤ U
1.2
for positive constants L and U. The unique determination of the optical coefficients is studied
in electrical impedance tomography problem 2–5 and some elliptic problem 6, which is
applicable to diffuse optical tomography problem also. Let us denote x μa , κ and Φ Φx to emphasize the dependence of Φ on the optical coefficient x.
Assuming we know some a priori information x0 about the structural optical coefficients x and the perturbation of the optical coefficients δx x − x0 , the diffuse optical
tomography problem is to find the perturbation of the optical coefficients δx from the difference Φx δx − Φx between the perturbed and unperturbed photon density distribution
on the boundary ∂Ω. The relation between δx and Φx δx − Φx is given by the following
Born expansion 7, 8:
Φx δx − Φx R1 x, δx R2 x, δx2 · · · ,
1.3
where
R0 x Φx,
Ri δxi R δx, Ri−1 δxi−1 , f ,
i 1, 2, . . . ,
R δx, f Rμa δx, f Rκ δx, f ,
δμa η R ·, η f η dη,
Rμa δx, f Rκ δx, f 1.4
Ω
Ω
δκ η ∇R ·, η · ∇f η dη,
and R·, η is the Robin function for a source at η, which is the solution of 1.1 for the optical
coefficient x when q is the Dirac delta function. By definition of 1.4, the operator R and R1
are different in the following sense:
R1 δx R δx, R0 Rδx, Φ.
1.5
Let the perturbation of the coefficients be δx† when we neglect second-order terms and higher
in the Born expansion 1.3. We can then formulate the linearized diffuse optical tomography
problem to find δx† from the following equation, which is the first-order Born approximation:
R1 δx† Φx δx − Φx.
1.6
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This linearized diffuse optical tomography problem is simple to implement and widely used
9, 10.
In this paper, a new method, which is more accurate than the linearized method, will
be suggested 1.6, which is based on the second-order Born approximation. And the method
is faster than the full nonlinear method 11. Let the solution of the proposed method in this
paper be δxB , and let δx be sufficiently small. Then, the error for the linearized solution δx†
and the proposed solution δxB is given by
†
δx − δx ≤ C† δx2A ,
A
B
δx − δx ≤ CB δx3A ,
A
1.7a
1.7b
where A L∞ Ω × L∞ Ω and C† and CB are constants which are independent of δx. Hence,
the error of the proposed solution xB in 1.7b is of the order Oδx3A , which is higher than
the order of the error of the linearized solution x† , Oδx2A .
The detailed statement with proof will be proved in Section 2. Numerical algorithm
involving the detailed computation of the second-order term is given in Section 3. Numerical
implementation of the proposed method and the linearized method is given in Section 4, and
the conclusion of the paper is given in Section 5.
2. Error Analysis
Instead of solving linearized solution δx† in 1.6, we suggest the second order solution δxB
satisfying
2
R1 δxB Φx δx − Φx − R2 δx† ,
2.1
2
R1 δxB − δx† −R2 δx† .
2.2
or equivalently,
In this section, we analyze the error for the linearized solution δx† and the suggested solution
δxB .
Let B H 1 Ω; then, the operator R and Ri , i 1, 2, . . ., are considered to be the
i times
operators from A × B → B and Ai A × · · · × A → B, respectively, by the definition given
in 1.4. For the detailed explanation about the definitions of higher-order Fréchet derivative
in diffuse optical tomography and its relation to the Born expansion, see 7.
Proposition 2.1. Let Φ be the solution of 1.1 for the given optical coefficients μa , κ, source q, and
modulating frequency ω. Then one gets the following relation between the operators between R and
Ri , i 1, 2, . . .:
i
≤ RiA×B → B ΦB
2.3
R i
A →B
for i 1, 2, . . ..
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Proof. By the induction argument on i 1, 2, . . . and using 1.5, we get the following inequality:
1
R A→B
≤ RA×B → B ΦB ,
2.4
which is 2.3 for i 1. Suppose that 2.3 holds for i 1, 2, . . . , I − 1. Then we obtain
I
R δxI ≤ RA×B → B δxA RI−1 δxI−1 B
≤ RA×B → B δxA RI−1 AI−1
B
δxI−1
A
2.5
≤ RIA×B → B δxIA ΦB .
Using 2.5 and the definition of the operator norm · AI → B , we obtain 2.3 for i I.
Therefore, by the induction argument, we have proved 2.3 for i 1, 2, . . ..
By 7, RA×B → B is bounded, and thus Ri Ai → B , i 1, 2, . . ., are also bounded by
Proposition 2.1. Let us assume that there exists a bounded operator R1 † from B to A such
that R1 † R1 idA . R1 † is usually called the left inverse of R1 . Let us denote
δx :
δxA ,
Φx :
ΦxB ,
R :
RA×B → B ,
i i
, i 1, 2, . . . ,
R :
R i
2.6
A →B
1 † 1 †
R :
R B→A
,
for brevity.
Using Proposition 2.1 and the assumption on the left inverse, the main theorem of this
paper is given as follows.
Theorem 2.2. Assume that there exists R1 † such that R1 † R1 id and R1 † is bounded, and
let
δx ≤
1
.
2R
2.7a
Then,
†
δx − δx ≤ C† δx2 ,
B
δx − δx ≤ CB δx3 ,
2.7b
2.7c
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5
where
1 †
2
C† :
2
R R Φ,
2
3 2
C†
C†
†
†
†
C
1 .
C :
RC C R
4R
2R
2.8
B
Proof. By 1.3 and 1.6, we obtain
R1 δx† − δx R2 δx2 R3 δx3 · · · .
2.9
Therefore we arrive at 2.7b by the following inequality:
Rδx2 Φ
1 †
†
2
†
δx − δx ≤ R 1 − Rδx ≤ C δx .
2.10
From 2.7a and 2.7b, we obtain the following upper bound of δx† :
†
δx ≤ 1 C† δx δx ≤ 1 R1 RΦ δx.
2.11
Using 2.2 and 2.9, we obtain
2
R1 δx − δxB R2 δx2 − R2 δx† R3 δx3 R4 δx4 · · · .
2.12
The second-order term on the righthand side of 2.12 is analyzed as follows:
2
R2 δx2 − R2 δx† Rδx, Rδx, Φ − R δx† , R δx† , Φ
R δx, R δx − δx† , Φ R δx − δx† , R δx† , Φ .
2.13
From 2.12, we obtain
2 1 † 2
3
2
3
4
B
2
† 4
δx − δx ≤ R R δx − R δx R δx R δx · · · .
2.14
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I Compute the solution Φx and the Robin function Rx and its first
and second derivatives.
II Find δx† by solving R1 δx† Φx δx − Φx as in 1.6.
III Find δxΔ δxB − δx† by solving R1 δxΔ −R2 δx† as in 2.2.
IV Compute δxB by adding δx† and δxΔ .
Algorithm 1: Numerical algorithm continuous version.
By using 2.3, 2.10, 2.11, 2.13, and the definition of C† , 2.7c is achieved from 2.14 as
follows:
Rδx3
1 † 2
B
†
†
δx − δx ≤ R Φ R δx − δx δx δx 1 − Rδx
†
1 †
ΦR2 δx3 C† 2 R1 RΦ 2R
R
≤
≤ C† δx3 C† 1 †
C
4R
2.15
R
≤ CB δx3 .
3. Numerical Algorithm
Assume that we can measure the photon density distribution Φxδx and Φx on the entire
boundary ∂Ω. That is to say, we have infinite detectors and one source. Then, the numerical
algorithm is given as follows.
The detailed computation of the integral operators R1 and R2 , which is introduced in
1.5, is as follows:
R1 δx Rμa δμa , Φ Rκ δκ, Φ,
R2 δx Rμa δμa , Rμa δμa , Φ Rμa δμa , Rκ δκ, Φ ,
Rκ δκ, Rμa δμa , Φ Rκ δκ, Rκ δκ, Φ.
3.1a
3.1b
3.1. Discretization
Algorithm 1 is based on one source and infinite detectors. However, for practical reasons, we
need to discretize Algorithm 1 to obtain the numerical algorithm for finite sources and finite
detectors for finite frequencies. The following notations will be used for the discretization:
i Nd detector positions: rid for id 1, 2, . . . , Nd ,
ii Ns source functions: qis δis Dirac delta function for is 1, 2, · · · , Ns ,
iii Nω frequencies: ωiω for iω 1, 2, . . . , Nω ,
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iv Ne elements: Tie for ie 1, 2, . . . , Ne ,
v Nn nodes: tin for in 1, 2, . . . , Nn ,
vi the measurement index: j iω − 1Ns Nd is − 1Nd id ,
vii the optical coefficient index: k iμκ − 1Ne ie , where iμκ is 1 the absorption
coefficient or 2 the diffusion coefficient.
If we use piecewise linear or bilinear finite element method, the finite element solution
is represented by
uh x Nn
uh in φin x,
3.2
in 1
where φin is the piecewise linear or the bilinear function which is 1 on the in th node and 0 on
all the other nodes. Assume μa and κ are piecewise constant function, which is constant for
each Ne finite elements. Therefore, in diffuse optical tomography inverse problem, we have
Nω Ns Nd measurement information contents and 2Ne variables to find.
We should discretize R1 and R2 to obtain a discretized version of Algorithm 1. Let the
Jacobian and Hessian matrices, which is the discretization of integral operators R1 and R2 , be
J and H. The relation between higher order derivatives for the diffusion operator and higher
order terms of Born expansions including R1 and R2 is analyzed in 7.
Firstly, let us discretize δx, Φ, and the Robin function R as follows:
δx ≈
N
e
δμie χTie ,
ie 1
Φiω ,is ≈
Nn
Ne
δκie χTie
,
3.3a
ie 1
Φiiωn ,is φin ,
3.3b
in 1
Riω ·, ris ≈
Nn
Riiωn ,is φin .
3.3c
in 1
Since we chose the source function qs as the Dirac delta function at the is th source point,
Φiω ,is Riω ·, ris . However, we will discriminate these two functions in this paper, since they
are different for general source function q which is different from the Dirac delta function. We
will use δμ instead of δμa for notational convenience.
Let the vector γ0 which corresponds to the discretization of δx in 3.3a be defined as
γ0 δμ1 , δμ2 , . . . , δμNe , δκ1 , δκ2 , . . . , δκNe .
3.4
By the adjoint method 12, Riω rid , · Riω ·, rid ∗ , where ∗ denotes complex conjugate.
Likewise for 3.3a, let γ, γ † , γ Δ , and γ B be the discretization of δx, δx† , δxΔ , and δxB ,
respectively.
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Journal of Applied Mathematics
For a function f and a measurable set T , let us denote f ∈ T if the intersection of the
support of f and T is not empty. The discretization of the linearized solution γ † is attained by
solving the following equation:
Jγ † b,
3.5
where
J j, k Riiωn1,id
φin1 ∈Tie φin2 ∈Tie
J j, k φin1 ∈Tie φin2 ∈Tie
∗
Eie in1 , in2 Φiiωn2,is
when iμκ 1,
∗
− 3 Riiωn1,id κ2ie Fie in1 , in2 Φiiωn2,is
when iμκ 2,
b j Φiω ,is x δxrid − Φiω ,is xrid ,
φin1 ξφin2 ξdξ,
Eie in1 , in2 3.6
Tie
∇φin1 ξ · ∇φin2 ξdξ.
Fie in1 , in2 Tie
The discretized solution γ Δ is obtained by solving the following equation:
t
Jγ Δ γ † Hγ † ,
3.7
where
H j, ie1 , ie2 φin1 ∈Tie1 φin2 ∈Tie2
Riiωn1,id
∗ Hμμ Hμκ Hκμ Hκκ ie1 , ie2 ; in1 , in2 Φiiωn2,is ,
3.8
where Hμμ , Hμκ , Hκμ , and Hκκ are the discretization of corresponding terms in 3.1b such
that
φin1 ξRiω ξ, η φin2 η dξdη,
Hμμ ie1 , ie2 ; in1 , in2 Tie1
Tie2
φin1 ξ∇η Riω ξ, η · ∇η φin2 η dξdη,
Hμκ ie1 , ie2 ; in1 , in2 Hκμ ie1 , ie2 ; in1 , in2 Tie1
Tie1 Tie2
∇ξ φin1 ξ · ∇ξ R
iω
ξ, η φin2 η dξdη,
∇ξ φin1 ξ · ∇ξ ∇η Riω ξ, η ∇η φin2 η dξdη.
Hκκ ie1 , ie2 ; in1 , in2 Tie1
Tie2
Tie2
3.9
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I Compute the solution Φγ0 iiωn ,is and the Robin function Rγ0 iiωd ,in for
iω 1, · · · , Nω , is 1, . . . , Ns , in 1, . . . , Nn as in 3.3b and 3.3c,
respectively.
II Find γ † by solving the equation 3.5.
III Find γ Δ by solving the equation 3.7.
IV Compute γ B by adding γ † and γ Δ .
Algorithm 2: Numerical algorithm discretized version.
Even though the Hessian is not discretized, we obtain the following discretized numerical
algorithm Algorithm 2, expecting the Hessian is simply discretized and approximated in
the next subsection:
3.2. Approximation of Hessian
In this subsection we approximate Hμμ , by assuming κ and μs are constant in Ω. The approximation is progressed in three ways.
First, we approximate the Robin function Rξ, η when ξ, η ∈ Ω \ ∂Ω by its leading
term R0 ξ, η defined by
⎧
2−p
1
⎪
⎪
ξ − η
⎪
⎪
κ
η
p
−
2
g
p
⎨
R0 ξ, η ⎪
⎪
1
2S
⎪
⎪
log ⎩
ξ − η
ω2 κ η
p ≥ 3,
3.10
p 2,
where gp is the hypersurface area of the unit sphere in Rp , p 2, 3, . . . and S supξ,η∈Ω |ξ − η|.
Some important relations between R and R0 are found in 13.
ie2 , the Robin function R and φin are approximated by constant
Second, when ie1 /
values R0 cie1 , cie2 and φin cie in Tie , respectively, where cie of the center of the ele
ie2 , 3.9 is approximated as follows:
ment Tie . That is to say, when ie1 /
Hμμ ie1 , ie2 ; in1 , in2 R0 cie1 , cie2 φin1 ξdξ
φin2 η dη.
3.11
Tie1
Tie2
Third, when ie1 ie2 , we use the following lemma.
Lemma 3.1. Let the measurable set T be contained in Rp , p 2, 3, . . ., and 0 < m < p; then, the following inequality holds for T :
1−m/p
m/p
ξ − η−m dξ dη ≤ p
gp |T |2−m/p , p ≥ 2,
p
−
m
T
1
4S2 π
2S
dξ dη ≤
1 log
log |T |2 , p 2,
ξ − η
4π
|T |
T
where |T | is the volume of T .
3.12a
3.12b
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Solution
Initial guess (0th approximation)
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1st approximation
2nd approximation
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
Figure 1: Jindex Nx ∗ Ny ∗ 0.4, Jalpha 6.4920e − 009, 10% noise, sources ∗, and detectors o.
Proof. If a ball with a radius r has the same volume as T , we have
r
p
|T |
gp
1/p
3.13
for the space dimensions p 2, 3, . . .. Let the ball of radius r with center ξ ∈ T be Bξ . Let
T0 T ∩ Bξ , T T \ Bξ , and T − Bξ \ T . Noting that |T | |T − |, we obtain
ξ − η−m dη T
ξ − η−m dη T0
ξ − η−m dη ≤
T0
r
T
T−
ξ − η−m dη
ξ − η−m dη gp p−m
≤
r
ρ
gp dρ p
−m
0
1−m/p
gp
p
≤
|T |
p−m
gp
p−m−1
ξ − η−m dη
Bξ
3.14
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Initial guess (0th approximation)
Solution
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1st approximation
2nd approximation
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
Figure 2: Jindex Nx ∗ Ny ∗ 0.4, Jalpha 6.5711e − 009, 10% noise, sources ∗, and detectors o.
for all ξ ∈ T . Therefore,
g |T |
ξ − η−m dηdξ ≤ p
p
−m
T
p
|T |
gp
1−m/p
p1−m/p m/p
gp |T |2−m/p .
p−m
3.15
Equation 3.12b is derived in the same manner.
Therefore, when ie1 ie2 , 3.9 is approximated using the inequality in Lemma 3.1 as
follows:
⎧
⎪
p2/p |Tie1 |12/p
⎪
⎪
⎪
⎪
⎨ 2 p − 2g 2/p κci e1
p
Hμμ ie1 , ie1 ; in1 , in2 ≈ φin1 cie1 φin2 cie1 ·
⎪
2
⎪
1
π
4S
⎪
⎪
⎪
|Tie1 |2
⎩ 8π 2 κci 1 log |T |
i
e1
e1
p ≥ 3,
p 2.
3.16
4. Numerical Implementation
In the numerical implementation, the following parameters are used:
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Initial guess (0th approximation)
Solution
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
1st approximation
2nd approximation
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
Figure 3: Jindex Nx ∗ Ny ∗ 0.3, Jalpha 1.8227e − 7, 10% noise, sources ∗ and detectors o.
i Ω 0, 6 × 0, 6 cm2 ,
ii Nd 16,
iii Ns 16,
iv Nω 1,
v Nx Ny 16,
vi Ne Nx ∗ Ny,
vii Nn Nx 1 ∗ Ny 1,
viii μa 0.05 0.2 − 0.05χD cm−1 ,
ix μs 8 cm−1 ,
x κ 1/3 ∗ μa μs 1/3 ∗ 0.05 8,
xi ω 2π ∗ 300 MHz,
xii a 1,
xiii Jindex Nx ∗ Ny ∗ 0.4.
Since the diffusion coefficient κ is constant, the right-hand side b is a Ns ∗ Nd column
vector, Jacobian J is a Ns ∗Nd ×Ne matrix, the Hessian H is a Ne ×Ns∗Nd×Ne third-order
tensor, and γ † t Hγ † is Ns ∗ Nd column vector in 3.5 and 3.7. H Hμμ is approximated
by 3.11 and 3.16.
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Initial guess (0th approximation)
Solution
1st approximation
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0.5
2nd approximation
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
Figure 4: Jindex Nx ∗ Ny ∗ 0.4, Jalpha 6.5029e − 009, 10% noise, sources ∗, and detectors o.
In the above setting, we reconstruct the obstacle D which has different absorption
coefficient 0.2 cm−1 compared to the background absorption coefficient 0.05 cm−1 . Four
cases of the obstacle D are considered in Figures 1, 2, 3, and 4. The reconstruction of the
absorption coefficient μa 0.05 0.2 − 0.05χD cm−1 is implemented using two algorithms.
One is the suggested Algorithm 2 based on the second-order Born approximation. The other
is linearized method based on the first-order Born approximation, which is equivalent to
the step I and II in Algorithm 2. We denoted these two methods in the figures: the 2nd
order approximation and the 1st-order approximation, respectively. On the upper-left part
of the figures, original μa and source/detector locations are plotted. The initial guess μa0
or γ0 for the absorption coefficient is plotted on the upper-right part of the figures. In the
lowerleft and lowerright part of each figure, reconstructed absorption coefficients by the first
†
approximation μa or γ † and the second approximation μBa or γ B are plotted, respectively.
In all four cases, 10% noise is added. Truncated singular value decompositionSVD
is used. Jindex is the number of largest singular values used in the truncated SVD method.
We used the Tikhonov regularization parameter Jalpha as the value of the Jindexth largest
singular values.
As is shown in the figure, the discrimination between background and the obstacle
is clearer in the second-order approximation than the first-order approximation. The reconstructed image resolution depends on the distance from the boundary of the tissue, which
is verified by comparing Figures 1 and 2 with Figures 3 and 4. And the resolution also depends on the size of obstacle, which is verified by comparing Figures 1 and 3 with Figures 2
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Journal of Applied Mathematics
and 4. Due to the diffusion property of near infrared light, the reconstructed image is much
blurred especially in Figure 3. The sensitivity to the noise made some kind of irregular checkerboard pattern near the boundary Figures 1, 3, and 4.
5. Conclusions
We derived a new numerical method based on the second-order Born approximation. The
method is a method of order 3, which is more accurate than the well-known linearized method based on the first-order Born approximation. The error analysis for the method is proved,
and the computation of the second-order term is explained using some approximation and
integral inequalities. The comparison between the suggested and the linearized method is
implemented for four different kinds of absorption coefficients. In the implementation, the
suggested method shows more discrimination between the optical obstacle and the background than the linearized method. If more accurate numerical quadrature with more
efficient approximation of the Robin function is used, the efficiency of the present method will
be elaborated. The simultaneous reconstruction of the absorption and the reduced scattering
coefficients based on the proper approximation on the second derivatives of the Robin
function would be an interesting topic.
Acknowledgment
This research was supported by Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education, Science and Technology 2010-0004047.
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